Question: The arithmetic mean of an odd number of consecutive odd integers is $y$. Find the sum of the smallest and largest of the integers in terms of $y$.
Solution: Let the first odd integer be $a$.  Let the rest of the odd integers be $a+2, a+4, a+6, \dots , a+ 2(n-1)$, for a total of $n$ integers.  The arithmetic mean of these integers is equal to their sum divided by the number of integers, so we have  \[ y = \frac{na + (2+4+6+\dots + 2(n-1))}{n}\] Notice that $2+4+6+\dots + 2(n-1) = 2(1+2+3+\dots + n-1) = 2\frac{(n-1)(n-1+1)}{2} = n(n-1)$.  Substituting and multiplying both sides by $n$ yields \[ yn = na + n(n-1)\] Dividing both sides by $n$, we have  \[ y = a+ n-1\] The sum of the smallest and largest integers is $a + a+ 2(n-1)$, or $2a+2(n-1)=2(a+n-1)=2y$.

Hence the answer is $\boxed{2y}$.